Design Analysis of Algorithm_Introduction-1.pptx

AD3351-DESIGN AND ANALYSIS
OF ALGORITHM
UNIT-1 INTRODUCTION

NOTION OF ALGORITHM
ALGORITHM
An algorithm is a sequence of unambiguous instructions for
solving a problem, i.e., for Obtaining a required output for any
legitimate input in a finite amount of time.

Some Important Points on Algorithm

CHARACTERISTICS OF AN
ALGORITHM
 Input - Valid inputs must be clearly specified
 Output – Produces at least one output- the result or solution of the problem
 Definiteness - Each step must be clear and unambiguously specified
Finiteness - Terminates after a finite number of steps
 Effectiveness - Steps must be simple enough to be performed using basic operations.

GCD(m,n) - Definition
 gcd(m,n) – the largest integer that divides both m and n evenly, ie., with
a remainder of zero.
 General method – gcd(m,n)
Eg: Compute GCD(60,24)
• Find common factors of 2 given numbers:
• Product of those common factors – gcd
=> gcd(24,18)=?

To illustrate the notion of the algorithm we consider 3
methods to solve the same problem – Computing the
greatest common divisor of 2 non-negative, not-both-
zero integers m,n => gcd(m,n)
 Euclid’s Algorithm
 Consecutive Integer Checking Algorithm
 Middle-School Algorithm

Compute gcd(60,24) using Euclid’s algorithm
Given: m=60 and n=24
Iteration 1:
n=24≠0 -> r=60 mod 24 -> r=12
Now, m=n=24 and n=r=12
Iteration 2:
n=12≠0 -> r=24 mod 12 -> r=0
Now, m=n=12 and n=r=0
Iteration 3:
n=0, condition fails and returns m value
Therefore, gcd(60,24)=12.

GCD(24,12)
Given: m=24 and n=12
Step 1: prime factors of m => 2,2,2,3
Step 2: prime factors of n => 2,2,3
Step 3: common factors => 2,2,3
Step 4: product of common factors => 2x2x3=12

Consecutive Integer Checking Algorithm

Contd..
Eg: (i) Compute gcd(9,6) using Consecutive Integer
Checking Algorithm
(i) Compute gcd(24,12) using Consecutive Integer
Checking Algorithm
(iii) Compute gcd(60,24) using Consecutive Integer
Checking Algorithm
Drawback:
Doesn’t work properly when one of its inputs is ZERO

Sieve of Eratosthenes
 Simple algorithm for generating consecutive primes not exceeding
any given integer n>1
 Starts by initializing a list of primes from 2 to n
 First Iteration – eliminate all multiples of 2 except 2
 Next Iteration - eliminate all multiples of 3 except 3
 Continuous until no more elements to be eliminated
 Remaining integers – primes

Algorithm : Sieve of Eratosthenes
Step1: Create a list of consecutive integers from 2 through n – (1,2,3,
…..,n).
Step 2: Initially let p=2 the smallest prime no.
Step 3: Enumerate the multiples of p by counting in increments of p from
2p to n and mark them in the list (these will be 2p,3p,4p,…, the p itself
should not be marked).
Step 4: Find the smallest no in the list greater than p that is not marked. If
there is no such no stop. Otherwise let p now equal to this new no and
repeat from Step 3.
Step 5: When the algorithm terminates the numbers remaining not marked
in the list are all the primes below n.

FUNDAMENTALS OF ALGORITHMIC PROBLEM SOLVING

Important Problem Types
 The different problem types are
1. Sorting
2. Searching
3. String processing
4. Graph problems
5. Combinatorial problems
6. Geometric problems
7. Numerical problems

Sorting
 Sorting problem is one which rearranges the items of a given list in
ascending order.
 For example in student’s information, we can choose to sort student records
in alphabetical order of names or by student register number. Such a
specially chosen piece of information is called a key.
 Sorting solves many problems, the most important of them is searching:
dictionaries, telephone books, class lists, and so on are sorted.

Contd..
 There are two important properties of sorting algorithms.
 A sorting algorithm is called stable, if it preserves the relative order of any two equal
elements in its input.
 For example, we have a list of students sorted alphabetically and if we sort the
student list based on their GPA: a stable algorithm will yield a list in which students
with same GPA will be sorted alphabetically.
 The second feature of an algorithm is ‘in place’, if it does not require extra memory.
 There are some sorting algorithms that are in place and those that are not.

Searching
 The searching problem deals with finding a given value, called a search key,
in a given set.
 The searching can be either a straightforward sequential search algorithm or
binary search algorithm based on different forms of representing the set.
 Some algorithms work faster but require more memory, some are very fast
but applicable only to sorted arrays.
 Searching, mainly deals with addition and deletion of records.

String processing
 A String is a sequence of characters. It is mainly used in string handling
algorithms.
 Most common ones are text strings, which consists of letters, numbers
and special characters.
 Bit strings consist of zeroes and ones and gene sequences, which can be
modeled by strings of four-character alphabet{A,C,G,T}

Graph Problems
 A graph is a collection of points called vertices,some of which are connected by
line segments called edges.
 Graphs can be used for modeling a wide variety of real-life applications such as
transportation, communication networks, project scheduling, and games.
 It includes graph traversal, shortest-path and topological sorting algorithms.
 The most well-known examples of computationally hard problems are the
traveling salesman problem and graph-coloring problem

Contd…
 The TSP is the problem of finding the shortest tour through n cities that visits
every city exactly once.
 It arises in modern applications as circuit board and VLSI chip fabrication, X-
ray crystallography and genetic engineering.
 The graph-coloring problem is to assign the smallest number of colors to
vertices of a graph so that no two adjacent vertices are of the same color.

Combinatorial Problems
 The traveling salesman problem and the graph-coloring problem are examples of
combinatorial problems.
 These are problems that ask us to find a combinatorial object such as
permutation, combination or a subset that satisfies certain constraints and has
some desired (e.g. maximizes a value or minimizes a cost).

Geometric Problems
 Geometric algorithms deal with geometric objects such as points, lines and
polygons.
 It also includes various geometric shapes such as triangles, circles etc.
 The applications for these algorithms are in computer graphic, robotics etc.
 The two problems most widely used are the closest-pair problems, given ‘n’ points
in the plane, find the closest pair among them.
 The convex-hull problem is to find the smallest convex polygon that would include
all the points of a given set.

Numerical Problems
 This is another large special area of applications, where the problems involve
mathematical objects of continuous nature: solving equations, computing definite
integrals and evaluating functions and so on.
 These problems can be solved only approximately.
 These require real numbers, which can be represented in a computer only
approximately.
 It can also lead to an accumulation of round-off errors.

Video Link
 https://youtu.be/fLYV-Z76dUM

Design Analysis of Algorithm_Introduction-1.pptx

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Design Analysis of Algorithm_Introduction-1.pptx